There's no hiding from the magickal associations of this name.
Rune (pronounced "ROON" or sometimes "ROO-neh") is an Old Norse name derived from run, meaning "secret" or "whisper." Runes are a popular method of divination today. If you're Neo-Pagan, you're going to hear about runes at some point.
Runes started out as a alphabet used by the Ancient Germanic peoples. The origins of this alphabet are uncertain, although the oldest known runic inscriptions date around AD 150. There are many different types of runic alphabets as the language evolved over time. But this was no ordinary alphabet. These letters would be used specifically for magickal charms and curses. There is no evidence that suggests that they were used for divination at this time. Due to the word's meaning, it's logical to assume that runes were a secret language only used by a certain class of people. Each rune can be an interesting inspiration for names, but I'll come back to that some other time.
During the Viking Revival of the 18th century, people became increasingly interested in runes, and the fascination hasn't faded. An Austrian mystic named Guido von List published Das Geheimnis der Runen ("The Secret of the Runes") in 1902, and the runes in this book would later capture the interest of the Nazis. J.R.R. Tolkien and J.K. Rowling were also inspired by runes, as they are mentioned or used in both of their books.
Runes are particularly important to Germanic Neo-Pagans. They are usually used for divination. Each rune is carved on a stone and they're laid out like tarot cards. Some groups also practice runic gymnastics in which they must mimic the shape of a particular rune with their bodies.
It's not clear when Rune became a given name, unless someone out there knows something that I don't. Rune is a popular name where Norse names are popular. It's rated #38 in Belgium and #374 in Norway. But in the United States, it's rare. I suppose that it could be used as a girls name, but I've never seen it suggested for girls.
Rune is one of my favorite boys names. It suggests someone who's pensive and strong. It seems like a name that would be popular with Neo-Pagans, but in practice I've been seeing a lot more Rowans, and really no Runes. I would love to see it used more in the namescape.